nLab genus

this entry is about the notion of genus in algebraic topology/cohomology. For classification of surfaces see instead the (related) entry genus of a surface, genus of a curve. There is also genus of a lattice.

Idea

Basic

For $R$ a (commutative) ring, an $R$-valued genus is a ring homomorphism into $R$

$$\sigma : \Omega_\bullet \to R$$

from a cobordism ring for cobordisms with specified structure; typical choices being orientation or stable complex structure. Often the rationalization of such a morphism is meant, see below at Properties – Rationalization.

To emphasize that this is indeed a ring homomorphism and hence in particular respects the multiplicative structure, a genus is sometimes (especially in older literature) synonymously called a multiplicative genus.

In stable homotopy theory and generalized cohomology theory

Since the cobordism ring is the ring of coefficients of the corresponding universal Thom spectrum, e.g $M U$, $M SO$, so a genus may also be written as a ring homomorphism of the form

$$M SO_\ast \longrightarrow R$$

or

$$M U_\ast \longrightarrow R$$

respectively. Written this way it is immediate that genera arise naturally as the value on homotopy groups (the “decategorification” or “de-homotopification”) of homomorphisms of E-∞ ring spectra from an actual universal Thom spectrum to some E-∞ ring $E$ with coefficient ring $R$

$$M SO \longrightarrow E$$

or

$$M U \longrightarrow E \,.$$

This in turn induces multiplicative morphisms of the cohomology theories represented by these spectra (the domain being hence cobordism cohomology theory), and these multiplicative maps are the “families version” of the given genus/index (Hopkins 94, section 3).

Such homomorphisms in turn arise naturally from universal orientations in generalized E-cohomology. Namely such an orientation is a homotopy of the form

$$\array{ \ast && \longleftarrow && B SO \\ & {}_{\mathllap{0}}\searrow & \swArrow_\sigma & \swarrow_{\mathrlap{J}} \\ && B GL_1(E) \\ && \downarrow \\ && E Mod }$$

(a trivialization of the $E$-(∞,1)-module bundle associated to the spherical fibration given by the J-homomorphism) and under forming homotopy colimits in $E$(∞,1)Mod this becomes an $E$-linear map

$$M SO \wedge E \longleftarrow E$$

hence a map

$$M SO \longleftarrow E \,.$$

At least in some important cases, genera seem to be naturally understood as encoding sigma-model quantum field theories. For $G$ some structure, the Thom spectrum $M G$ is the classifying space of manifolds with G-structure, and hence may be thought of as classifying target spaces for sigma-models. The codomain spectrum $R$ itself may then be thought of as a classifying space for a certain class of QFTs, and hence the genus $\sigma : M G \to R$ can be thought of as assigning to any target space the corresponding sigma-model.

This is for instance the case at least over the point for the A-hat genus $M Spin \to K O$, which may be thought of as sending manifolds with spin structure to the corresponding (1,1)-supersymmetric EFT (“spinning particle”); and for the Witten genus $M String \to tmf$, which can be thought of as sending a manifold with string structure to the corresponding (2,1)-supersymmetric EFT (“heterotic string”).

Properties

Rationalization

When the coefficient ring $R$ does not have additive torsion, then any ring homomorphism

$$\phi \colon \Omega^{U/SO}_\bullet \longrightarrow R$$

is determined already by its rationalization

$$\phi \colon \Omega^{U,SO}_\bullet\otimes \mathbb{Q} \longrightarrow R \otimes \mathbb{Q}$$

which is traditionally denoted by the same symbol. The rational cobordism rings in turn are known to be polynomial rings

$$\Omega^U_\bullet\otimes \mathbb{Q} \simeq \mathbb{Q}[\mathbb{C}P^1,\mathbb{C}P^2, \cdots ]$$

$$\Omega^{SO}_\bullet\otimes \mathbb{Q} \simeq \mathbb{Q}[\mathbb{C}P^2,\mathbb{C}P^4, \cdots ]$$

whose generators are identified with the cobordism classes of the manifolds which are the complex projective spaces, as indicated.

Logarithm and Characteristic series

Definition in components

Given a (rational) genus $\phi \colon \Omega^{U,SU}_\bullet\otimes \mathbb{Q} \to R \otimes \mathbb{Q}$ one defines (we follow (Hopkins 94))

1. its logarithm to be the formal power series over $R \otimes \mathbb{Q}$ given by

   $$log_\phi(z) \coloneqq \sum_{n \in \mathbb{N}} \phi(\mathbb{C}P^n) \frac{z^{n+1}}{n+1};$$

2. its characteristic series (or Hirzebruch series) to be the formal power series over $R \otimes \mathbb{Q}$

   $$K_\phi(z) \coloneqq \frac{z}{\exp_\phi(z)} \,,$$

   where $\exp_\phi$ is the inverse of the logarithm;

3. its characteristic class as the universal characteristic class which via the splitting principle is fixed by its value on the universal line bundle as

   $$K_\phi(c_1) \in H^\bullet(B U(1),R \otimes \mathbb{Q})$$

   where $c_1 \in H^2(B U(1), \mathbb{Z})$ denotes the universal first Chern class; hence its value on a direct sum $L_1 \oplus \cdots \oplus L_k$ of complex line bundles is

   $$\prod_{i} K_\phi(c_1(L_i)) \,.$$

Definition via orientations in generalized cohomology

Suppose that the given genus $\Omega_\bullet^{SO} \longrightarrow R$ indeed comes from an orientation in generalized cohomology (as discussed above) hence from a homomorphism of E-∞ rings

$$\beta \;\colon\; M SO \longrightarrow E$$

for an E-∞ ring $E$ with homotopy groups $R\simeq \pi_\bullet(E)$. (And suppose that $E$ defines a complex oriented cohomology theory.)

This defines (Ando-Hopkins-Rezk 10, prop. 2.11) a universal orientation of real vector bundles and hence of complex vector bundles and hence of complex line bundles in $E$-cohomology

$$\beta \;\colon\; M U \longrightarrow M SO \stackrel{\beta}{\longrightarrow} E \,.$$

Now rationally, i.e. for $E\otimes \mathbb{Q}$, there is a canonical such orientation, given by the composite

$$\alpha \;\colon\; M U \longrightarrow M SO \longrightarrow H \mathbb{Q} \simeq \mathbb{S}\otimes \mathbb{Q} \stackrel{}{\longrightarrow} E \otimes \mathbb{Q} \,.$$

Thus, given any orientation $\beta$, its rationalization may be compared to $\alpha$. Since these rational orientations are equivalently trivializations of maps to $B GL_1(E \otimes \mathbb{R})$, their difference is a class $\beta/\alpha$ with coefficients in $GL_1(E\otimes R)$, hence over any space $X$ the difference is a class in $H^0(X, \pi_\bullet E\otimes \mathbb{Q})^\times$.

Specifically consider the delooping $X= B U(1)$ of the circle group. For this the cohomology ring is the power series ring in a single variable (the universal first Chern class $c_1(L)$). Under the canonical inclusion $B U(1)\to B U$ both the above orientations $\beta$ and $\alpha$ pull back, so that we have a difference

$$K_\beta \coloneqq \beta/\alpha \in (E_\bullet \otimes \mathbb{Q})[ [ c_1(L) ] ] \,.$$

This is the Hirzebruch series of $\beta$ (Ando-Hopkins-Rezk 10, def. 3.10).

If $F$ denotes the formal group law classified via $MU_\bullet \to M SO_\bullet \stackrel{\beta_\bullet}{\to} E_\bullet$ then

$$K_\beta(z) = \frac{z}{\exp_F(z)}$$

The Hirzebruch formula

The central theorem of (Hirzebruch 66) expresses the genus of an arbitrary (cobordism class of a) manifold $X$ of dimension $2n$ via the formula

$$\phi(X) = \langle \prod_{i = 1}^n K_\phi(x_i(T X)), [X] \rangle$$

in terms of the Hirzebruch characteristic series $K_\phi$ discussed above, and via the splitting principle:

This means that $\prod_{i = 1}^n K_\phi(x_i(T X))$ is the function of Chern classes $c_k$ (i.e. Pontryagin classes $P_{2k}$ and Euler classes $\chi$) obtained by rewriting the polynomial in the $x_i$ (the “Chern roots”) as a polynomial in elementary symmetric polynomials $\sigma_k(x_1,\cdots, x_n)$ and then substituting for each of these by $c_k(T X)$.

(see also e.g. ManifoldAtlas – Genera – 4.1 Construction).

Examples

Todd genus

The Todd genus is the genus with logarithm

$$-log_{Todd}(1-x) = \sum_{n \in \mathbb{N}} \frac{x^n}{n}$$

Signature genus

The signature genus;

$\hat A$-genus

The A-hat genus is the index of a Dirac operator coming from a spin bundle in KO-theory. It is given by the characteristic series

The characteristic series of the $\hat A$-genus is

$$\begin{aligned} K_{\hat A}(e) & = \frac{z}{e^{z/2} - e^{-z/2}} \\ &= \exp\left( - \sum_{k \geq 2} \frac{B_k}{k} \frac{z^k}{k!} \right) \end{aligned} \,,$$

where $B_k$ is the $k$th Bernoulli number (Ando-Hopkins-Rezk 10, prop. 10.2).

The $\hat A$-genus is an integer on manifolds with spin structure.

Elliptic genus

An elliptic genus is one whose logarithm is given by

$$log_{ell}(z) = \int_0^z (1-2 \delta t^2 + \epsilon t^4)^{-1/2} d t$$

for constants $\delta, \epsilon$ with non-degenerate values $\delta^2 \neq \epsilon$ and $\epsilon = 0$.

For degenerate choices this reproduces the signature genus and the A-hat genus above, see at elliptic genus for more. For non-degenerate values one may regard $\epsilon$ and $\delta$ as values of modular forms of the same name and hence regard all elliptic genera together as one single genus with coefficients in $MF_\bullet(\Gamma_0(2))$. This “universal” elliptic genus is the Witten genus.

Witten genus

The Witten genus

$$w \colon \Omega^{SO}_\bullet \longrightarrow \mathbb{Q}[ [q] ]$$

is the genus with coefficients in the power series ring $\mathbb{Q}[ [ q ] ]$ with characteristic series given by

$$\begin{aligned} K_w(z) & = \frac{z/2}{sinh(z/2)} \prod_{n \geq 1} \frac{(1-q^n)^2}{(1-q^n e^z)(1-q^n e^{-z})} \\ & = \exp\left( \sum_{k \geq 2} G_k \frac{z^k}{k!} \right) \end{aligned} \,,$$

where $G_k$ are the Eisenstein series (Ando-Hopkins-Strickland 01, Ando-Hopkins-Rezk 10, prop. 10.9). (Notice that the constant term in $G_k$ is proportional to the $k$th Bernoulli number, so that indeed the exponential expression matches that for the A-hat genus above.)

On manifolds with spin structure the Witten genus takes values in $\mathbb{Z}[ [ q ] ]$

On manifolds with rational string structure it takes values in (the $q$-expansion of) modular forms for $SL_2(\mathbb{Z})$, meaning that setting $q = e^{2 \pi i \tau}$ then as a function $f$ of the parameter $\tau$ taking values in the upper half plane the Witten genus satisfies

$$f(-1/\tau) = (-\tau)^n f(\tau) \,.$$

Finally on manifolds with actual string structure it takes values in topological modular forms. See at Witten genus for more.

Non-examples

Euler characteristic

The Euler characteristic $X \mapsto \chi(X)$ is close to being a genus, but is not cobordism invariant

(this is the index of the Dirac operator $D = d + d^\dagger$)

Related concepts

partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:

$d$	partition function in $d$-dimensional QFT	supercharge	index in cohomology theory	genus	logarithmic coefficients of Hirzebruch series
0			push-forward in ordinary cohomology: integration of differential forms			orientation
1	spinning particle	Dirac operator	KO-theory index	A-hat genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin \to KO$
	endpoint of 2d Poisson-Chern-Simons theory string	Spin^c Dirac operator twisted by prequantum line bundle	space of quantum states of boundary phase space/Poisson manifold	Todd genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
	endpoint of type II superstring	Spin^c Dirac operator twisted by Chan-Paton gauge field	D-brane charge	Todd genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
2	type II superstring	Dirac-Ramond operator	superstring partition function in NS-R sector	Ochanine elliptic genus		SO orientation of elliptic cohomology
	heterotic superstring	Dirac-Ramond operator	superstring partition function	Witten genus	Eisenstein series	string orientation of tmf
	self-dual string		M5-brane charge
3						w4-orientation of EO(2)-theory

References

The abstract concept of genus is due to Friedrich Hirzebruch. It had evolved out of the older concept of (arithmetic) genus of a surface via the concept of Todd genus introduced in

• John Arthur Todd, The arithmetical invariants of algebraic loci, Proc. London Math. Soc. (2), Ser. 43, 1937,

  190–225.

An review of the history is at the beginning of (Hirzebruch-Kreck 09)

The theory of multiplicative sequences and characteristic series of genera is due to

• Friedrich Hirzebruch, Neue topologische Methoden in der algebraischen Geometrie, Ergebnisse der Mathematik und ihrer Grenzgebiete 9, Springer (1956) [doi:10.1007/978-3-662-41083-7]

• Friedrich Hirzebruch, Topological methods in algebraic geometry, Grundlehren 131, Springer (1966) [doi:10.1007/978-3-642-62018-8, MR0202713 (34 #2573) Zbl 0843.14009]

Early review:

• Friedrich Hirzebruch, Thomas Berger, Rainer Jung, chapter 2 of: Manifolds and Modular Forms, Aspects of Mathematics 20, Viehweg (1992), Springer (1994) [doi:10.1007/978-3-663-10726-2, pdf]

Further review:

• Serge Ochanine, What is… an elliptic genus?, Notices of the AMS, volume 56, number 6 (2009) (pdf)

• Friedrich Hirzebruch, Matthias Kreck, On the concept of genus in topology and complex analysis, Notices of the AMS, volume 56, number 6 (2009) pdf

• Michael Hopkins, section 2 of Topological modular forms, the Witten Genus, and the theorem of the cube, Proceedings of the International Congress of Mathematics, Zürich 1994 (pdf)

• Manifold Atlas, Formal group laws and genera

• Wikipedia, Genus of a multiplicative series

Discussion in terms of orientations in generalized cohomology and specifically for the A-hat genus and the Witten genus is in

• Matthew Ando, Michael Hopkins, Neil Strickland, Elliptic spectra, the Witten genus and the theorem of the cube, Invent. Math. 146 (2001) 595–687 MR1869850

• Matthew Ando, Mike Hopkins, Charles Rezk, Multiplicative orientations of KO-theory and the spectrum of topological modular forms, 2010 (pdf)